30
1 A Historical Review of the Structures of Water and Ice
an H 3 O
+ ion can be transferred by the Grotthuss mechanism: the transferring proton
is not necessarily the same one which brought this charge from the previous step of
the diffusion.
Thus, the charge transfer is carried out through a proton “sticking” from H 3 O
+
to H 2 O, as shown in Fig. 1.16a. In this way, δr = 0. That is why a hydronium ion,
or proton, always has the same diffusion coefficient as an oxygen atom, because the
proton does not jump, but sticks from one molecular species to another, just like a
relay race, where the baton has the same mobility as the runner. This mechanism
assumes a finite lifetime of H 2 O molecules, which varies from 10 h [77, 78] to
ms [79], depending on the initial assumptions and the model.
1.5 Diffusion by Neutron Scattering
Quasi-elastic neutron scattering is a relatively new technique [80]. The first accurate
data on water appeared in the 1970s when Powles and Page measure the mixtures
of heavy and light water [81, 82]. But, the first results on water were even earlier;
Singwi in the 1960s suggested using neutrons for the study of the diffusive motion of
water molecules [83]. When neutrons interact with hydrogen atoms, there could be
an exchange of momentum and energy, which depends on the diffusive motion of the
water molecules. Since neutrons interact with protons only, we can track hydrogen
atoms, but cannot say anything about diffusion of oxygen atoms. In this context,
neutron scattering complements X-ray diffraction (see Sect. 1.2). Note, however,
that as we saw above, hydrogen and oxygen atoms are not necessarily attached to
each other, even if they have the same diffusion coefficient (see Sect. 1.4).
Figure 1.23 shows a typical quasi-elastic neutron scattering spectra for water.
Scattering intensity is plotted as a function of the energy transfer E. The natural
line of water appears as a result of the subtraction of an experimentally measured
curve by the spectrometer resolution function. Figure 1.24 shows the half-width
of the corresponding natural quasi-elastic scattering line for water as a function of
wave vector κ. This curve contains all the information on the diffusion of protons.
But, in order to extract this information, one again needs an initial assumption on
atomic-molecular dynamics, and a preliminary understanding of the type (rotations,
translations, drift, etc.) of molecular motion in water.
The Bernal–Fowler model of water (see Sect. 1.3.1) assumes that diffusion occurs
by an interstitial mechanism through the large displacements of molecules from a
stationary vibrational state to another quasi-equilibrium state. This idea continues
in modern quantum-chemical calculations, which show that molecular diffusion in
water goes through so-called wait-and-switch dynamics [85]. The approach assumes
that water molecules have an oscillatory-diffusional motion with a diffusion coefficient D of ≈ 10
−9 m
2 /s, periodically lingering for a time t 0 ≈ 4 ps in localized
vibrational states. The main difference for the Bernal–Fowler approach is that the
oscillatory state is not stationary, because the center of oscillations drifts with time,
obeying Brownian motion. Figure 1.25a sketches the trajectory of a proton that corre-
1 A Historical Review of the Structures of Water and Ice
an H 3 O
+ ion can be transferred by the Grotthuss mechanism: the transferring proton
is not necessarily the same one which brought this charge from the previous step of
the diffusion.
Thus, the charge transfer is carried out through a proton “sticking” from H 3 O
+
to H 2 O, as shown in Fig. 1.16a. In this way, δr = 0. That is why a hydronium ion,
or proton, always has the same diffusion coefficient as an oxygen atom, because the
proton does not jump, but sticks from one molecular species to another, just like a
relay race, where the baton has the same mobility as the runner. This mechanism
assumes a finite lifetime of H 2 O molecules, which varies from 10 h [77, 78] to
ms [79], depending on the initial assumptions and the model.
1.5 Diffusion by Neutron Scattering
Quasi-elastic neutron scattering is a relatively new technique [80]. The first accurate
data on water appeared in the 1970s when Powles and Page measure the mixtures
of heavy and light water [81, 82]. But, the first results on water were even earlier;
Singwi in the 1960s suggested using neutrons for the study of the diffusive motion of
water molecules [83]. When neutrons interact with hydrogen atoms, there could be
an exchange of momentum and energy, which depends on the diffusive motion of the
water molecules. Since neutrons interact with protons only, we can track hydrogen
atoms, but cannot say anything about diffusion of oxygen atoms. In this context,
neutron scattering complements X-ray diffraction (see Sect. 1.2). Note, however,
that as we saw above, hydrogen and oxygen atoms are not necessarily attached to
each other, even if they have the same diffusion coefficient (see Sect. 1.4).
Figure 1.23 shows a typical quasi-elastic neutron scattering spectra for water.
Scattering intensity is plotted as a function of the energy transfer E. The natural
line of water appears as a result of the subtraction of an experimentally measured
curve by the spectrometer resolution function. Figure 1.24 shows the half-width
of the corresponding natural quasi-elastic scattering line for water as a function of
wave vector κ. This curve contains all the information on the diffusion of protons.
But, in order to extract this information, one again needs an initial assumption on
atomic-molecular dynamics, and a preliminary understanding of the type (rotations,
translations, drift, etc.) of molecular motion in water.
The Bernal–Fowler model of water (see Sect. 1.3.1) assumes that diffusion occurs
by an interstitial mechanism through the large displacements of molecules from a
stationary vibrational state to another quasi-equilibrium state. This idea continues
in modern quantum-chemical calculations, which show that molecular diffusion in
water goes through so-called wait-and-switch dynamics [85]. The approach assumes
that water molecules have an oscillatory-diffusional motion with a diffusion coefficient D of ≈ 10
−9 m
2 /s, periodically lingering for a time t 0 ≈ 4 ps in localized
vibrational states. The main difference for the Bernal–Fowler approach is that the
oscillatory state is not stationary, because the center of oscillations drifts with time,
obeying Brownian motion. Figure 1.25a sketches the trajectory of a proton that corre-